Calibration and Model Simulation Results

My parameter choices are given in Table 1.13. I have opted for smaller average jump sizes with an average disaster size of 15%. Barro (2006) uses the dataset of Madison (2003) and found the average disaster size to be 29%. Barro and Ursua (2008) update Madison (2003) and find that the average disaster size is between 21-22%; this disaster distribution is also used in Wachter (2013).34 I opt for smaller average disaster sizes in line with evidence from Nakamura, Steinsson, Barro, and Ursua (2013), who document partial recoveries after disasters, and estimate the average permanent impact of disasters to be about 15%. While the actual probability of these smaller disasters is 5.85%, I opt for a more conservative calibration of 4%, which is achieved using a ¯ξ = 0.0355 as in Wachter (2013) along with an average jump size of µλ = 0.03 in the event of a jump in λt. Figure 1.8 compares my

multinomial jump size distribution with smaller average jump sizes to the distribution used in Barro and Ursua (2008) and Wachter (2013). An important challenge in calibrating representative investor models is to match the high observed volatility of the price-dividend ratio. The model places an upper bound on the amount of volatility in the state variables that can be allowed for solutions to exist (this is clearly seen in the equations for the Epstein-Zin discount factor in Appendix A.1.5). I fix σξ such that the discriminant in the

solution to bξ is zero, which helps match the high volatility of the price-dividend ratio and

also reduces the number of free parameters. Theλtprocess is calibrated to be less persistent

volatility, low Sharpe ratio, and low lease rate. The model-implied gold lease rate is 0.93%, which compares well to the 1% lease rate in the data. Lease rates in the model are the convenience yield, which corresponds to the dividend yield (dividend over price). For com- parison, I also present the model 90% confidence intervals for simulation paths in which no disaster occurred. While these no-disaster intervals are more appropriate to compare against stock and bond moments (since no disasters have occurred in the recent U.S. data, on which the stock and bond returns are based), for gold and platinum returns it is more natural to compare against population moments, since there have been 21 economic dis- asters from 1975 - 2006 in international markets (using my disaster cutoff) based on the Barro and Ursua (2008) dataset (including several OECD countries), which can conceiv- ably affect gold and platinum returns and volatilities. The model explains the expected returns, volatilities and lease rates for platinum as well, including the high lease rate and high volatility. The model also accounts for time variation in GP, with the volatility and persistence of GP falling right inside the 90% confidence intervals. The median persistence for all simulations matches the data estimate nearly perfectly. Following this, I run the below return predictability regressions using model excess stock returns and GP:

1 h

h

X

i=1

log(R_te_+i)−log(Rb_t+i) =β0+β1log(GPt) +t+h.

The left hand side is the normalized excess return for one year up through five years ahead, while the right hand side is the model GP. The results are shown in the top panel of Table 1.15. The data estimates fall right in the model confidence intervals, with the data R2 estimates very close to the median values.35 Thus, the model can explain the observed predictability of returns by GP. Similar to the data, the model delivers very low to negligible dividend growth predictability, similar to (Wachter (2013)). The model can also account for the observed relationship between GP and the slope of the implied volatility curve for index options, as detailed in Appendix A.1.7.

35_{It is difficult to decide which, all simulations or no disasters, is most appropriate for the predictability} exercise, since U.S. stock returns were not affected by domestic disasters, while the GP ratio is potentially affected by international disasters. For completeness, I include both sets of results.

How well have I captured the effect of supply dynamics on GP? Is there predictability coming from the supply effects (including autocatalyst demand)? The second panel of Table 1.15 investigates this issue. I regress GP on logZt inside the model, and we see that the data

estimate falls right inside the 90% interval.36 Since the leading coefficient on logZt in the

model depends on 1, this serves as a further check on the assumed complementarity ( <1) between jewellery (gold and platinum) and nondurable consumption. Under a calibration where >1 as in Barro and Misra (2013), this regression in the data results in a coefficient smaller than 1. The second regression in this panel investigates return predictability by logZt in both the model and the data. In the model, logZt does not predict returns by

construction, although in small samples it is occasionally possible to spuriously find weak evidence of predictability. Both the population and median values, however, show that there is no predictability coming from the supply channels. I run the same regression in the data and find no evidence of predictability through logZt, which is further evidence that

the predictability does not come from a cash flow channel.

These results for repeated samples of 39 years lead to an interesting finding. Time-variation in GP over finite samples is affected by logZt, which is not a priced variable in this economy.

The third panel shows return predictability regressions where I control for the effect of logZt,

which adds volatility and persistence to GP without adding predictive power. We see in this case that the point estimates increase at all horizons, and now the 90% interval for return predictability by GP does not contain 0. The R2 increase over all horizons quite dramatically. In the data, we can separately identify logZt and logAt, the aggregate per-

capita stock of platinum used as autocatalysts. Empirically, a regression of GP on logZt

for persistent supply effects, the persistence of GP is lower; the monthy AR(1) coefficient is 0.962, which implies a half-life of about 1.5 years.